A generalization of the $q$-Lidstone series

In this paper, we study the existence of solutions for the general $q$-Lidstone problem:

$$(D_{q^{-1}}^{r_n}f)(1)=a_n,(D_{q^{-1}}^{s_n}f)(0)=b_n,(n\in \mathbb{N})$$

where $(r_n{)}_n$ and $(s_n{)}_n$ are two sequences of non-negative integers and $(a_n{)}_n$ and $(b_n{)}_n$ are two sequences of complex numbers. We define a $q^{-1}$-standard set of polynomials and then we introduce a generalization of the $q$-Lidstone expansion theorem.